Has it been proven that, if $\ y_n = x_{n+1} - x_n\ $ is non-decreasing, then $\ x_n\ $ cannot be a counter-example to Erdős Conjecture?

I'm trying to find a subset $\ A\ $ of $\ \mathbb{N}\ $ that disproves Erdős Conjecture on Arithmetic progressions.

If we instead write $\ A\ $ as a (strictly) increasing sequence of integers, $\ (x_n)_n.\ $ My question is, has it been proven that, if $\ y_n = x_{n+1} - x_n\ $ is non-decreasing, then $\ x_n\ $ cannot be a counter-example to Erdős Conjecture? In particular, I was thinking about such sequences with the further property that there are infinitely many $\ n\in\mathbb{N}\ $ such that $\ x_n \geq 2 x_{n-1},\ $ because then at least we can be sure that these $\ x_n\ $ do not make arithmetic progressions of length three with any two $\ x_{k_1},\ x_{k_2} \ $ if $\ k_1,k_2 < n.$

I cannot think of how to disprove that such sequences (with or without the further property) can be both a "large set", and also have a maximum length arithmetic progression, but maybe there is a relatively simple proof, possibly using the pigeonhole principle, or maybe this hasn't been proven yet?